What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras...) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; the Lie bracket being the Poisson bracket $\{ \cdot , \cdot \}$.

The symplectic form induces a surjective Lie algebra homomorphism between $(C^\infty(M), \{ \cdot , \cdot \} )$ and the hamiltonian vector fields on $M$, which is a Lie subalgebra of the Lie algebra of vector fields. Therefore the properties of $(C^\infty(M), \{ \cdot , \cdot \} )$ are related to the Lie algebra of vector fields on $M$.

Are there any references (in English) on infinite dimensional Lie algebras treating the examples above?

Any reference dealing with a specific symplectic manifold will be very useful (specially to rule out general statements).

Kac's book on Infinite dimensional Lie algebras deals with Kac-Moody algebras, and "E. Cartan, Les groups de transformations continus, infinis, simples, C. R. Acad. Sc., t.144 (1907) 1094." is in French (I cannot read it).